J. Phys. B: Atom. Molec. Phys., 1971, Vol. 4.Printed in Great Britain
WKB approximation and threshold law for
electron-atom ionization
R. PETERKOP
Physics Institute, Latvian Academy of Sciences, Riga, Salaspils, USSR
MS. received 23rd ,Voz.mabev 1970
Abstract. The solutions of the Hamilton-Jacobi equation and of the continuity equation for electron-atom ionization problem at zero total angular
momentum are investigated in the neighbourhood of saddle points of the
potential. The solutions are found which describe sets of orbits whose points
of condensation coincide with the saddle points of the potential. The asymptotic behaviour of solutions corresponds to a spherically outgoing wave or t o
a plane wave. The energy dependence of WKB wave functions is determined
and by means of a matching procedure the Wannier ionization threshold law
has been confirmed. The connection between the WKB approximation and
the asymptotic expansion of the zero-energy wave function has been considered.
1. Introduction
The WKB approximation was applied earlier (Peterkop and Liepinsh 1969) to a
simplified problem of ionization in which electrons move in one dimension. T h e
non-linear threshold law obtained by Wannier (1953) was confirmed for that case.
I t may be concluded on physical grounds that the one-dimensional model should
give the correct threshold behaviour of the total ionization cross section. The
physical reason is the Coulomb repulsion because of which the low-energy electrons
leave the atom in opposite directions. However, the behaviour of the differential
ionization cross section as well as the asymptotic form of the wave function can be
investigated only with the help of the three-dimensional WKB approximation.
T h e wave function in the WKB approximation has the form
where S is the solution of the Hamilton-Jacobi equation
(O1S)2+ ( V p y
=
2(E-
v-)
and P is the solution of the continuity equation
Vl(PV1S) + V 2 ( P V 2 S )
0.
(3)
If S and P are real they can be interpreted as the classical action and the classical
density.
W e use units in which the mass and charge of the electron are equal to unity. T h e
nucleus is taken to be infinitely heavy. I n the general case the wave function in the
WKB approximation can be written as a linear combination of functions of the
type (1). W e shall consider the simplest case of motion when the total angular
momentum is equal to zero. T h e functions S and P depend then on three coordinates
p =
=
e = c0s-1(P,i2).
(Y12+Y22)1’2
513
(4)
R. Peterkop
5 14
Equations (2) and (3) become
where
Z ( X 6')
, =
1
__
cos tl
1
+sin c(
l
1
(1 -cos 0 s i n 2 ~ ) ~ l ~
a
__(fsin2 2 ~ )
sin2 2cr act
Dlf
=
D2f
= -___-
a
4
(f sin Q).
sin2 2 x sin Q aQ
(9)
The function Z(K,Q) in the form (7) satisfies the case of ionization of the hydrogen
atom.
2. Solution of the Hamilton-Jacobi equation
Wannier (1953) showed that the most important region for ionization at low
energies is at r1 = - r2 which agrees with the neighbourhood of the saddle point of
the function Z(a,0). Therefore, we shall investigate the solutions of ( 5 ) and (6)
at tl N ~ / and
4 Q N T.
We expand Z ( R ,Q) in a series
Z
where
=
Z~+~Z,(Acl)2++-Z2(AQ)2+
...
AV. = % - ~ / 4
AQ = Q - r .
I t follows from (7) that
I n accordance with the expansion (lo), we seek the solution of the HamiltonJacobi equation in the form
S = S&) + + S ~ ( ~ ) ( A K ) ~ + & S ~ ( ~ ) ...
( A.Q ) ~ +
Substituting (10) and (13) into ( 5 ) we obtain equations
dS0
- - w
dSi
dP
Si2
2,
dP
P2
P
w-+-=-
where
i = 1,2
(13)
Threshold law foy electron-atom ionization
515
T h e solutions are
So = p w
+ -In
2 0
P(X+WI2
X
220
(17)
where
C i j are arbitrary constants.
The integration constant for So is chosen so that for E
=
0
so= (8Z0p)1’2.
It follows from (12) and ( 2 3 ) that pl is real and p 2 is imaginary
mll is negative and m12is positive
m,,
=
mI2 = 1.127.
-1.627
The function u l j is real but uZ3;is complex. Since uZ1 =
(26)
u2 is real if C21 = Cz2*.
3. Orbit equations
T h e physical meaning of the function S is determined by the set of orbits which
it describes. T h e orbit equations are
d ri
-=
vis
d.~~
t
where t is the time.
Using coordinates (4), (27) takes the form
Substituting (13, 14, 18), taking into account only the first terms of the series obtained and eliminating t we get
Hence it follows
d% A%du,
- = -__
da
Accdu,
-=
dP
U1
dP
=
dP
%(P)
-_.
U2
dP
A8 = u2(p).
R. Peterkop
516
T h e hypergeometric function in (21) can be expressed by the Legendre function.
Then (30) coincides with the formulae for orbits found by Wannier (1953).
Since S , contains the function U, in the form of a logarithmic derivative, S ,
has only one arbitrary constant which can be chosen as the ratio C1,/C12 s c,.
Analogously, S, is dependent only on the ratio C21/C22= c2. Thus, the function S
has two arbitrary constants. T h e equations (30) also follow from
as
-=
ac,
as
_- const.
const
ac,
T h e derivatives may be calculated making use of the Wronskian
dui,
dui,
dp
dP
U . - - U .12 __ =
pi(2Zo)1'2
p2"
T h e constants c, and c2 determine the points RI and R, for which u,(Ri)= 0.
I n the general case, R iis complex and may be multivalued. For fixed c, and c p ,
equations (30) describe a set of orbits for which R, and R, are the condensation points
(intersection points). Thus, the solutions of the Hamilton-Jacobi equation in the
form (13-23) describe all possible sets of orbits whose centres or condensation points
are located at tl 77/4, 0 = 77 but arbitrary p.
T h e function Sihas a pole at p = R,. I n the vicinity of the pole, expansion (13)
can be applied if A x or A0 is small enough.
=T
4. Asymptotic behaviour
With the help of the formulae of the analytic continuation of hypergeometric
function (Erdelyi 1953) we find for Ep + CO
(
2Ep
(33)
2,
where
r ( x ) denotes the gamma function.
T h e asymptotic behaviour differs for the cases when the condensation points are
located at a finite distance or at infinity.
If RI # CO which means Cilyil+ Ci2yi2# 0, then ui const. and (30) describes
a spherically symmetric set of outgoing orbits. Equation (33) then leads to
N
1
S
N
...
xp+ ~ { Z o + k Z , ( A a ) 2 + ~ Z p ( A 0 ) 2 ~ l n p +
X
(35)
which, with account taken of (lo), confirms the asymptotic form obtained earlier
(Peterkop 1962, Rudge and Seaton 1965).
If R, = CO and R, = CO, then ut w constlp and
S
N
xp(l AX)' - i(A0)')
+ ... .
(36)
Here allowance for S, and S2 changes the leading term in the asymptotic form of S.
Threshold law foy electron-atom ionization
517
I n this case Av. and A0 tend to zero if p -+ CO. I n the asymptotic region all the orbits
have the same direction y. = 7714,0 = m-, A set of parallel straight lines corresponds
to a plane wave. However, for the complete construction of the plane wave, it is
necessary to take into account the dependence on other coordinates. T h e function
which corresponds only to zero angular momentum will not suffice.
T h e classical action corresponding to a plane wave is
I n Wannier's case k ,
S = k,r,+ k,r,,
- k,. Then 12, = X/1,/2 and
=
=
(37)
xp(i- i(ay.)z
- g(ae)z+ ...Icos q5
(38)
where q5 denotes the angle between k , and r , - r,.
It is seen that the Ay. and A b dependence in formula (36) corresponds to (38).
5. Zera-energy case
At E = 0 we have
Uij =
p"!.
T h e function So is determined by (24), and for i
(39)
=
1, 2
For p --f CO, the function u12 i
m and the remaining u i j + O . Hence R, = CO if
c1 = CO. At c, = 0 we have R, = 0 which corresponds to a set of orbits coming
(with respect to E ) from the centre of coordinates. However, with respect to 0, no
similar set of orbits exist. At any c2, u2( CO) = 0. If C,, = C2,*, then u2 is real, it
oscillates and has an infinite number of zeros.
Formula (40) is simplified in the extreme cases where c, and c, are equal to zero
or infinity. Then S becomes
A
=
(8.Z0)1'2(1+ -(Ax),+
mli
4
--(Ab)2+...
m2j
16
Substituting (41) into ( 5 ) we obtain the equation for A
Equations (41) and (43) were obtained earlier by Rudge and Seaton (1965).
6. The density function
Analogously to (13) the solution of (6) may be expanded as
P
=
Po(p)+ terms involving
+ ... ,
(44)
R.Peterkop
518
We restrict ourselves to finding Po. Substitution of (13) and (44) into (6) leads to
Po =
C
(45 1
p5W241ZC,2
where C is an arbitrary constant.
We consider solutions corresponding to the spherically outgoing waves, that is
the cases when R, and R, are finite. Introducing the normalization condition
we obtain
Po 3 p - 5
as
p 3 m
c = X(CIlY11+ C12Y12)(C,lY,, + c,2Y,2)2.
I t follows from (34) that for E + 0
C
-
C12E1-m12,
(46)
(47)
(45)
7. Threshold behaviour of ionization cross section
The threshold law can be determined with the help of a procedure which matches
the exact wave function with an approximate one for which the energy dependence
is known.
We assume that the WKB approximation can be applied at p 2 Y,,. Then in this
region
~ j =' fpol12exp
(' S )
(49)
where f is the matching coefficient.
We choose c, such that I?, # a. Then with the help of (35) and (46) we obtain
for p -f m
T h e differential ionization cross section is
where KO is the momentum of the incident electron.
The wave function at a finite distance should be finite for E -+ 0
U?(i~,,>
+const.
as
E
-to
Then it follows from (49) that
and with the help of (48) we get
Ud(7i/4,7i) N E m l , - 1 / 2 .
(54)
Formula (54) describes the threshold behaviour of the differential ionization
cross section at CI = ~ / and
4 0 = T . The total ionization cross section is
utot = 2x2
ud(%,0) sin2 ZCIsin 0 dcr.de.
(55)
Threshold law for electron-atom ionization
519
I t bas shown by Vinkalns and Gailitis (1967) that the classical ionization cross section
is practically constant with respect to y. but has a sharp maximum at 8 = T . From
(30) and (33, 34) it is seen that, for fixed C,, and CZ2,the final values of A6 (i.e., the
values at p -+ CO) decrease as El 4. For E + O , the orbits condense to 8 = n which
was the keynote of Wannier's theory. I t should be noted that the condition A K 2: 0
for ionization orbits is fulfilled only in the vicinity of the nucleus. I n the asymptotic
region a can be arbitrary. The behaviour of A0 is opposite: A8 N 0 in the asymptotic region while in the vicinity of the nucleus it can be arbitrary.
We can approximate ud with the following model
u d z , 8) = 4 7 7 / 4 ,
Ilel
for
for
=o
lA8Imax = const. El
6
lA8jmax
IAOI >
IAolmax
4.
(56a)
( 5 6b)
(57)
Substitution of ( 5 6 , 57) into (55) leads to
utot
N
E1'2ud(7r/4,5 5 )
(58)
N
which confirms Wannier's result.
In the matching equation (49), the WKB functions may be used which correspond
to different (but finite) values of R,,as well as linear combinations of such functions.
But as (48) is valid for all such functions they all lead to (54) and (58).
T h e peculiarity in the dependence on 8 of the differential cross section was not
noted by Rau (1971) who solved the Schrodinger equation using an approximation
similar to the WKB approximation. With the help of some additional assumptions
he obtained the energy dependence of the differential cross section without the term
- 4 in the exponent of (54).
8. Connection with the asymptotic expansion of the wave function
For E = 0, the solution of the Schrodinger equation corresponding to the quasiclassical cases when ci is equal to zero or infinity can be expanded in an asymptotic
power series in 2 / p .
T h e Schrodinger equation is
where A satisfies (43) we obtain the recursion relations
=
ae
T h e WKB approximation is obtained by putting +=
i 0 in (61). T h e left hand
side of (61) follows from the continuity equation (6) if one substitutes in it
S
=
Adp
p
= b,Zp-9!2-m-n
(62)
520
R. Peterkop
Since b-, = 0, bo is the same in the quantum and quasiclassical cases. Hence
the WKB approximation is valid at sufficiently large p when the sum in (60) can be
replaced by its first term.
T h e quantity m in the exponent in (60) cannot be arbitrary. Substituting (42)
into (61) and assuming that bo is neither singular nor zero we obtain
m
=
m,i+2m2j
(63)
which agrees with (39) and (45).
There exist 4 different functions A determined by (42,43) and differing in the
choice of m,, and m Z j . T h e use of m2, leads to an exponentially increasing wave
function if A8 # 0. T h e choice of m,, gives the asymptotic form of a plane wave,
with respect to v.. Thus, to describe the asymptotic form of the ionization wave
function, we can use only m,, and mZ2which corrresponds to c, = c2 = 0. For
p -+ CO, the wave function decreases exponentially if A8 # 0.
It should be noted that S , is real if jczl = 1. I n this case the WKB wave function
does not decrease or increase exponentially. However, because of the oscillations of
the function u2 it will have an infinite number of poles.
Equation (42) implies the 2Al2v. = aAj80 = 0 for x = 7714 and 0 = T . However (43) has also solutions with finite derivatives at these values of v. and 8. For
a simplified problem, it is possible to show that such solutions determine sets of
orbits which in the asymptotic region form a plane plus ingoing (or outgoing) wave.
If Z( x , 0) = Z,, then (43) has a solution
A
%-Ea,
=
(8z0)1’2c0s--2
where v., can be arbitrary.
On the other hand, equation (5) in the simplified case has solutions (Peterkop
1969)
As‘,’ = xp cos (G! - v.,) + CD
(65)
$ 2 ) = xp-@
(66)
where
CD = Qq(y- 1)+ -In x77(1+Y)2
X
42,
zo
S1)
and S2)determine two branches of the same hyperbolic orbit.
of S‘,) on
K
T h e dependence
in the asymptotic region corresponds to a plane wave which has direction
EO.
For E
=
0, we obtain
S‘1’
%--Eo
- S‘2’ =
(8ZOP)1’2cos 2
(69)
which agrees with (64).
References
ERDELYI,
A . , 1953, Higher Transcendental Functions, Vol. 1 (New York: McGraw-Hill), chap. 2.
PETERKOP,
R., 1962, Zh. eksp. teor. Fiz., 43, 616-8 (Sov. Phys.-JETP, 16, 442-4, 1963).
- 1969, Abstr. 6th Int. Conf. on the Physics of Electronic and Atomic Collisions (Cambridge
Mass.: M.I.T. Press), pp. 936-9.
Threshold law for electron-atom ionization
521
PETERKOP,
R., and LIEPINSH,
A., 1969, Abstr. 6th Int. Conf.on the Physics of Electronic and
Atomic Collisions (Cambridge, Mass: M.I.T. Press), pp. 212-4.
RAU,A. R. P., 1971, Phys. Rec., in press.
RUDGE,M. R. H., and SEATOX,
M. J., 1965, Proc. R. SOC.
A, 283, 262-90.
VINKALNS,
I., and GAILITIS,
&I., 1967, Abstr. 5th Int. Conf. on the Physics of Electronic and
Atomic Collisions (Leningrad : Nauka), pp. 648-50.
WANNIER,
G. H., 1953, Phys. Rea., 90, 817-25.